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数学代写 - MATH4210 Financial Mathematics Assignment 2

时间：2020-10-12

THE CHINESE UNIVERSITY OF HONG KONG

Department of Mathematics

MATH4210 Financial Mathematics 2020-2021 T1

Assignment 2

Due date: 16 October 2020 11:59 p.m.

Please submit this assignment on blackboard. If you have any questions

regarding this assignment, please email your TA Wong Wing Hong

(whwong@math.cuhk.edu.hk).

1. Suppose the continuous compounding interest rate is r and the price of

a stock is S(t) at time t. If it pays dividend d×S(tD) at time tD, where

0 < tD < T and 0 < d < 1, show that its forward price F (0, T ) satisfies

F (0, T ) =

1

1 + d

S(0)erT under no arbitrage opportunity assumption.

2. (Put-Call Parity Relation with Dividend) Assume that the value of the

dividends of the stock paid during [t, T ] is a deterministic constant D

at time tD ∈ (t, T ]. Let S(t) be the stock price, r be the continuous

compounding interest rate, CE(t,K) and PE(t,K) be the prices of Eu-

ropean call and put option at time t with strike K and maturity T

respectively. Show that

CE(t,K)− PE(t,K) = S(t)−Ke−r(T−t) −De−r(tD−t)

for all t < T .

3. Assume that the value of the dividends of the stock paid during [t, T ]

is a deterministic constant D at time tD ∈ (t, T ]. Let S(t) be the stock

price, r be the continuous compounding interest rate, CA(t,K) and

PA(t,K) be the prices of American call and put option at time t with

strike K and maturity T respectively. Show that

CA(t,K)− PA(t,K) < S(t)−Ke−r(T−t)

for all t < T .

4. Suppose the continuous compounding interest rate is r, two European

call options has same strike K and different maturity T1 < T2. Suppose

2the underlying asset pays a deterministic dividend D at tD ∈ (T1, T2].

Prove

CE(t, T1) < CE(t, T2) + (De

−r(tD−t) −K(e−r(T1−t) − e−r(T2−t)))+

for all t < T1.

5. Suppose that we have the following 3 European call with the same

maturity T in the financial market:

Type Strike Price Price at time 0

Call 90 15

Call 100 12

Call 110 5

Suppose that the continuous compounding interest rate is r = 0 in the

market and the maturity time is T = 1. Can you construct an arbitrage

portfolio with the above options?